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Mirrors > Home > MPE Home > Th. List > Mathboxes > erbr3b | Structured version Visualization version GIF version |
Description: Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.) |
Ref | Expression |
---|---|
erbr3b | ⊢ ((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 786 | . . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝑅 Er 𝑋) | |
2 | simplr 788 | . . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐵) | |
3 | simpr 476 | . . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐶) | |
4 | 1, 2, 3 | ertr3d 7647 | . 2 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐵𝑅𝐶) |
5 | simpll 786 | . . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝑅 Er 𝑋) | |
6 | simplr 788 | . . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵) | |
7 | simpr 476 | . . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶) | |
8 | 5, 6, 7 | ertrd 7645 | . 2 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) |
9 | 4, 8 | impbida 873 | 1 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 class class class wbr 4583 Er wer 7626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-er 7629 |
This theorem is referenced by: (None) |
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